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Euclidian metric spaces

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Euclidian metric spaces are 1,2,3,…,n dimensional, where n is a natural number. In this article Euclidian metric spaces are extended to spaces with a complex number of dimensions or even non number at all. One can insert in place of n natural, complex or another type of quantity of dimensions and obtain the corresponding type of Euclidian metric space. For example 1.5 dimensional space. Something between line and plane.

Let there be an open, connected region in complex plane which includes points 1 and . Let there be a set of functions defined in the region . Let there be the following norm, distance and scalar product defined for the functions of the set  :

where

More details one can see at www.oddmaths.info.

Summation in the case of analytical functions is taken with the Caves summation formula for indefinite sum:

Summation

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Indefinite sum

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where and periodical function with the period one


where is a parameter, are Bernoulli numbers and



The Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle {\emph {floor}}} of ( is real) is the largest integer less then . The boundaries of summation are determined for example from the folloving condition

or where is a constant.

are chosen the least that satisfy the inequality.

Definite sum is defined as:

More details one can see at www.oddmaths.info/indefinitesum.

Summation of non-analytical functions

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Let there be a set of functions such that a streams to zero when streams to infinity faster then any power of the inverse of , i.e. for any . The set is the space of basic functions. Let there on the space of basic functions be defined a functional

The functional of finite difference of a function is defined as follows:

where

Definition of the functional of the sum of a function .

A function belongs to the space of basic functions . First I define the functional of sum on the functions . From the previous result therefore

where is an indefinite sum of . For the rest functions I choose

Therefore the functional is defined on the entire space

Examples

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Heaviside function of the second type and Dirac delta function of the second type

and

or their shifted forms

and

Summation with non-number boundaries

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Let zero matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. TeX parse error: Undefined control sequence \emph"): {\displaystyle \ \,{\emph {0}}\ ,} and identity matrix are {} matrices. is {} orthonormal matrix with orthonormal vectors and . Let , where is Hermitian conjugate of matrix and

then by definition

If then

--Ascoldcaves (talk) 00:37, 3 November 2011 (UTC)